Let 
be a triangle with circumcircle 
and incentre 
. A line 
intersects the lines 
, 
, and 
at points 
, 
, and 
, respectively, distinct from the points 
, 
, 
, and . The perpendicular bisectors 
, 
, and 
of the segments 
, 
, and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to 
.
Let $ABC$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$.
The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$.
Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
Školjka